High School

If [tex]f(4) = 246.4[/tex] when [tex]r = 0.04[/tex] for the function [tex]f(t) = P e^{rt}[/tex], then what is the approximate value of [tex]P[/tex]?

A. 50
B. 210
C. 289
D. 1220

Answer :

To solve for the value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{rt} \)[/tex], use the given information [tex]\( f(4) = 246.4 \)[/tex] when [tex]\( r = 0.04 \)[/tex].

1. First, write down the equation with the known values:
[tex]\[
f(4) = P \cdot e^{0.04 \times 4}
\][/tex]

2. Substitute the given value for [tex]\( f(4) \)[/tex]:
[tex]\[
246.4 = P \cdot e^{0.16}
\][/tex]

3. Calculate [tex]\( e^{0.16} \)[/tex]. From a calculation, we know:
[tex]\[
e^{0.16} \approx 1.1735
\][/tex]

4. Substitute this value back into the equation:
[tex]\[
246.4 = P \cdot 1.1735
\][/tex]

5. Solve for [tex]\( P \)[/tex] by dividing both sides by [tex]\( 1.1735 \)[/tex]:
[tex]\[
P \approx \frac{246.4}{1.1735} \approx 209.97
\][/tex]

6. Round [tex]\( P \)[/tex] to the nearest whole number for the multiple-choice answer:
[tex]\[
P \approx 210
\][/tex]

Therefore, the approximate value of [tex]\( P \)[/tex] is 210. The correct answer is B. 210.